Subsets and intervals

A is a subset of B, written A ⊂ B, if every element of A is also in B. With B = {1, 2, 3, 4}: {2, 3} ⊂ B, but {1, 5} ⊄ B, because 5 is not in B.

• Every set is a subset of itself: A ⊂ A.
• φ is a subset of every set (it has no element that could fail the test).
• If A ⊂ B and B ⊂ A, then A = B.
• If A ⊂ B, then B is a superset of A.

Counting subsets: a set with n elements has 2ⁿ subsets. The subsets other than the set itself are proper subsets: there are 2ⁿ – 1 of them. A set of 4 elements has 16 subsets, 15 of them proper.

Example: one set has 48 more subsets than another. Then 2ᵐ – 2ⁿ = 48 = 16 × 3 = 2⁴(2² – 1), so 2ᵐ = 64 and 2ⁿ = 16: the sets have 6 and 4 elements.

The power set P(A) is the set of all subsets of A, so it has 2ⁿ elements. For A = {p, q}, P(A) = {φ, {p}, {q}, {p, q}}. Even P(φ) = {φ} is not empty: it has one element, φ.

Element or subset? Use ∈ for a single element and ⊂ for a set made of elements. For A = {1, {2, 3}}:

• 1 ∈ A and {2, 3} ∈ A: both are elements.
• {1} ⊂ A and {{2, 3}} ⊂ A: each is a set built from elements of A.
• But {2, 3} ⊄ A, because 2 on its own is not an element of A.

The number sets sit inside one another: N ⊂ W ⊂ Z ⊂ Q ⊂ R.

Intervals are subsets of R. A square bracket includes the end point, a round one leaves it out:

So {x ∈ R : –2 < x ≤ 5} = (–2, 5], and [0, 3) = {x ∈ R : 0 ≤ x < 3}. Each has length b – a.

A universal set U contains all the sets in a particular discussion. For {2, 4}, {3} and {2, 5, 6}, the set {1, 2, ..., 10} can serve as U; so could all of N. U depends on the problem.